The topic for the next few weeks will be Riemann surfaces. First, however, I need to briefly review harmonic functions because I will be talking about harmonic forms. I will have more to say about them later, and I actually won’t use most of today’s post even until then. But it’s fun.

Some of this material has also been covered by hilbertthm90 at A Mind for Madness.

**Definition**

A function on an open subset of , , is called **harmonic** if it satisfies the **Laplace equation** For now, we are primarily interested in the case , and we will identify with . In this case, as is well-known, harmonic functions are locally the real parts of holomorphic functions.

**The Poisson Integral**

The following fact is well-known: given a continuous function on the circle , there is a unique continuous function on the closed unit disk which is harmonic in the interior and coincides with on the boundary.The idea of the proof is that can be represented as a **Fourier series**,

where the are obtained through the orthogonality relations

where the inner product is the product taken with respect to the Haar measure on the circle group. This convergence holds in , because the exponentials form an orthonormal basis for that space. Indeed, orthonormality can be checked by integration, and the Stone-Weierstrass theorem implies their linear combinations are dense in the space of continuous functions on the circle. It is even the case that convergence holds uniformly if is well-behaved (say, ). But this is only for motivational purposes, and I refer anyone interested to, say, Zygmund’s book on trigonometric series for a whole lot fo such results.

Now, it is clear that the functions

are harmonic (where ) as the real parts of .

It thus makes sense to define the extended function as (more…)